The Topology of Tile Invariants

TL;DR

This paper employs topological methods, specifically 2-complexes and spherical pictures, to analyze tile invariants and their algebraic structures, providing new insights into planar tiling problems.

Contribution

It introduces a topological framework linking tile counting groups to second homology groups of 2-complexes, enhancing understanding of tile invariants.

Findings

Tile counting group is isomorphic to a quotient of second homology group.

Derived well-known tile invariants using topological techniques.

Applied the framework to solve a tiling problem involving modified rectangles.

Abstract

In this note we use techniques in the topology of 2-complexes to recast some tools that have arisen in the study of planar tiling questions. With spherical pictures we show that the tile counting group associated to a set $T$ of tiles and a set of regions tileable by $T$ is isomorphic to a quotient of the second homology group of a 2-complex built from $T$. In this topological setting we derive some well-known tile invariants, one of which we apply to the solution of a tiling question involving modified rectangles.